A wave automaton for Maxwell's equations

This paper presents an extension to electromagnetic fields of the wave automaton, which was introduced in recent years for describing wave propagation in inhomogeneous media. Using elementary processes obeying a discrete Huygens' principle and satisfying fundamental symmetries such as time reversal and reciprocity, this new wave automaton is capable of modeling Maxwell's equations in 3+1 dimensions. It supplements the methods that were developed early for scalar and spinor fields. Received 19 July 2001     

Maxwell's equations in Segal's model: Solutions and their invariance

We determine all smooth solutions of Maxwell's equation in Segal's universe; furthermore we show that the group of diffeomorphisms stabilizing the space of solutions of these equations is the conformal group of Segal's model.Aspirant du F.N.R.S.     

Exact solutions of Einstein and Einstein-scalar equations in 2 + 1 dimensions

A nonstatic and circularly symmetric exact solution of the Einstein equations (with a cosmological constant Λ and null fluid) in 2 + 1 dimensions is given. This is a nonstatic generalization of the uncharged spinless BTZ metric. For Λ = 0, the spacetime is though not flat, the Kretschmann invariant vanishes. The energy, momentum, and power output for this metric are obtained. Further a static and circularly symmetric exact solution of the Einsteinmassless scalar equations is given, which has a curvature singularity atr = 0 and the scalar field diverges atr = 0 as well as at infinity.     

Euclidean gauge fields in four dimensions from stochastic differential equations

Using a stochastic differential equation for field configurations in a three-dimensional space lattice, one proves existence of a euclidean theory for non-abelian gauge fields on the lattice. The theory is also shown to possess a scaling limit for the mass gap at weak coupling. The construction is extended in a less rigorous manner to a theory with fermions and gauge fields.     

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schr?dinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.     

Bäcklund transformations for nonlinear evolution equations in 2 + 1 dimensions

We obtain the general form of the Bäcklund transformations for the nonlinear evolution equations solvable through the Zakharov and Shabat method and, as an example, we write them down explicitly for the Kadomtsev-Petviashvili equation, the two-dimensional “three-wave” equation and the Davey-Stewartson equation.     

AnO(3, 1) approach to Maxwell's equations

Using the principal series representations of the Lorentz group, a method parallel to that of Gelfand and Yaglom is suggested to obtain Maxwell's equations, which dispenses with the arbitrary introduction of a degenerate transformation with respect to which the photon equations are invariant. The method also gives subsidiary conditions which, in conjunction with the masslessness of the particle, yield the Lorentz condition and the correct values of photon polarization.     

Shear free solutions of Maxwell's equations

A twisting shear free solution of Maxwell's equations is obtained by transforming to a complex coordinate system in which the corresponding solution is complex but twist free. The equations in this system are easily solved, and, by transforming back to the original coordinates, a twisting shear free solution of Maxwell's equations is obtained.     

A note on supersymmetry breaking in 1+1 dimensions

The spontaneous breakdown of supersymmetry for a single chiral superfield in 1+1 dimensions is investigated and a number of puzzling features are resolved by noting that the loop expansion is not always valid. It is found that both broken and unbroken supersymmetry are stable with respect to radiative corrections.     

Canonical structure of the CP sigma model in 1 + 2 dimensions

The CP^N model with a θ-term in 1 + 2 dimensions is quantized canonically. By employing a suitable parametrization we demonstrate that solitons carry θ-dependent fractional spins.